Time splitting is a numerical procedure used in solution of partial differential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiffness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly. In the present work a nonlinear advective equation is solved numerically using different techniques based on a generalised framework for splitting methods. The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their influence on the split solutions has been made. An understanding of influence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made. In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods. Three time-level schemes also need explicit filtering to remove the computational mode. This filtering can have a significant impact on the obtained numerical solutions, and hence three-time level schemes appear to be unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation.

La simulation d’écoulements incompressibles pose de nombreuses difficultés. Une première est la question de savoir comment traiter la contrainte d’incompressibilité et le couplage vitesse/pression afin d’obtenir une solution précise à moindre coût. Pour cela, nous nous intéressons en particulier à deux méthodes de time splitting : la correction de pression et la correction de vitesse. Une seconde difficulté porte sur des conditions limites de sortie. Nous nous intéressons ici à deux d’entre elles : la condition limite de traction et la condition limite d’Orlanski. Après avoir détaillé les difficultés pouvant apparaître lors de l’implémentation des méthodes de time-splitting, nous proposons une nouvelle implémentation de la condition limite de traction qui permet d’améliorer les ordres de convergence obtenus. Nous nous intéressons ensuite à la condition limite d’Orlanski qui nécessite une certaine vitesse d’advection C dans la direction normale à la limite dont nous proposons ici une nouvelle définition. Nos propositions sont confrontées à de multiples écoulements physiques afin de valider leurs comportements : l’écoulement en aval d’une marche descendante, l’écoulement au niveau d’une bifurcation,l’écoulement autour d’un obstacle et des écoulements de Poiseuille-Rayleigh-Bénard. / One of the understudied difficulties in the simulation of incompressible flows is how to treat the incompressibilityconstraint and the velocity/pressure coupling in order to obtain an accurate solution at low computationnalcost. In this context, we develop two methods: pressure-correction and velocity-correction. An anotherdifficulty is due to the boundary conditions. We study here two of them : the traction boundary condition andthe Orlanski boundary condition. After having developed the difficulties that appears when implementing timesplittingmethods, we propose a new way to enforce the traction boundary condition which improves the orderof convergence. Then we propose a new definition of the advective velocity C which is needed for the Orlanskiboundary condition. Our propositions are validated against multiple physical flows: flow over a backward facingstep, flow around a biffurcation, flow around an obstacle and several Poiseuille-Rayleigh-Bénard flows.